One of the intrinsic factors that affects the growth of two phytoplankton species is called the inter-competition coefficient. When this parameter value is decreased, the first phytoplankton specie benefit from biodiversity gain whereas the second phytoplankton specie is vulnerable to biodiversity loss. In contrast, when the same parameter value is increased from the value of 0.0525 to 0.099 the first phytoplankton specie dominantly suffers from a biodiversity loss whereas the second phytoplankton specie benefits from a biodiversity gain. The novel results that we have obtained have not been seen elsewhere but compliments our current contribution to knowledge in this challenging interdisciplinary research; these full results are presented and discussed quantitatively.

1. International Journal of Advanced Engineering, Management and Science (IJAEMS) [Vol-4, Issue-10, Oct-2018]
https://dx.doi.org/10.22161/ijaems.4.10.4 ISSN: 2454-1311
www.ijaems.com Page | 737
Modelling the Increasing Differential Effects of
the First Inter-Competition Coefficient on the
Biodiversity Value; Competition between two
Phytoplankton Species
P. Y. Igwe1, J. U. Atsu2, E.N. Ekaka-a3, and A.O. Nwaoburu3
1Department of Mathematics, Federal College of Education (Technical), Omoku, Rivers State, Nigeria.
2Department of Mathematics/Statistics, Cross River University of Technology, Calabar, Nigeria.
3Department of Mathematics, Rivers State University, Nkpolu,Port Harcourt, Nigeria.
Abstract— One of the intrinsic factors that affects the
growth of two phytoplankton species is called the inter-
competition coefficient. When this parameter value is
decreased, the first phytoplankton specie benefit from
biodiversity gain whereas the second phytoplankton specie
is vulnerable to biodiversity loss. In contrast, when the
same parameter value is increased from the value of 0.0525
to 0.099 the first phytoplankton specie dominantly suffers
from a biodiversity loss whereas the second phytoplankton
specie benefits from a biodiversity gain. The novel results
that we have obtained have not been seen elsewhere but
compliments our current contribution to knowledge in this
challenging interdisciplinary research; these full results are
presented and discussed quantitatively.
Keywords— Differential effect, inter-competition
coefficient, phytoplankton specie, biodiversity richness,
continuous differential equation.
I. INTRODUCTION
Heterogeneity of species is a very important factor in
biodiversity richness. Phytoplankton play a very crucial role
in ocean ecology [Saha and Bandyopadhyay 2009].
While some species of phytoplankton are known to produce
toxins which can contaminate seafood, others are high
producers of biomass which in high concentrations can
cause mortalities of marine life. However, the toxin
producing phytoplankton are known to play very significant
roles in the growth of zooplankton [Bandyopadhyay et al
2008]. Atsu & Ekaka-a (2017) have shown that as fractional
order dimension increases from 0.1 to0.75, there is a
reduction in specie depletion. The implication is that an
increased fractional order dimension results in a
biodiversity gain.
II. MATERIALS AND METHODS
We have considered the following Lotka–Volterra model
equations of competition indexed by a systemof continuous
non-linear first order ordinary differential equations:
𝑑𝑁1 (𝑡)
𝑑𝑡
=𝑁1(𝑡)[𝛼1 -𝛽1 𝑁1(𝑡) -𝛾1 𝑁2(𝑡)] (1)
𝑑𝑁2 (𝑡)
𝑑𝑡
=𝑁2(𝑡)[𝛼2 -𝛽2 𝑁2(𝑡) -𝛾2 𝑁1(𝑡)] (2)
Here, the initial conditions are defined by 𝑁1(0)=𝑁10 ≥ 0
and 𝑁2(0)=𝑁20 ≥ 0, whereas 𝑁1(𝑡) and 𝑁2(𝑡) specify the
densities of the two phytoplankton species (measured as the
number of cells per litter). For the purpose of this
formulation, 𝛼1and 𝛼2 specify the cell proliferation rate per
day;𝛽1 and 𝛽2 specify the rate of intra-specific competition
terms for the first and second species;𝛾1 and 𝛾2 specify the
rate of inter-specific competition. The units of 𝛼1, 𝛼2, 𝛽1,
𝛽2, 𝛾1and 𝛾2 are per day per cell and day is the unit of time.
Following the parameter values as proposed by
Bandyopadhyaya, et al ,2008, where 𝛼1 = 2, 𝛼2 = 1, , 𝛽1 =
0.07, 𝛽2 = 0.08, 𝛾1 = 0.05, 𝛾2 = 0.015.
The method is hereby stated step by step as follows:
Step 1: Consider a scenario where 𝑁1(old) is the predicted
biomass [𝑁1(old) is the population density of the first
phytoplankton species otherwise called the biomass of the
first phytoplankton species when all other parameter values
are fixed at time t.
Step 2: Replace 𝑁1(old) with 𝑁1(new) due to a variation of
the inter-competition coefficient 𝛾1.

2. International Journal of Advanced Engineering, Management and Science (IJAEMS) [Vol-4, Issue-10, Oct-2018]
https://dx.doi.org/10.22161/ijaems.4.10.4 ISSN: 2454-1311
www.ijaems.com Page | 738
Step 3: If 𝑁1(new) is strictly less than 𝑁1(old), it indicates
that the variation of 𝛾1has predicted a depletion which
mimics biodiversity loss. In this scenario, the appropriate
mathematically formula for the quantification of
biodiversity loss is defined as follows:
BL(%)= 100[
𝑁1
( 𝑜𝑙𝑑)− 𝑁1 (𝑛𝑒𝑤)
𝑁1(𝑜𝑙𝑑)
]
Step 4: If 𝑁1(new) is strictly greater than 𝑁1(old), due to the
variation of 𝛾1,then a biodiversity gain has occurred which
can be similarly defined as follows:
BG(%)=100 [1-
N1(new)
N1(old)
]
III. RESULTS
On the application of the above mentioned methods, we
have obtained the following empirical results that we have
not seen elsewhere.
Table.1: Evaluating the extent of biodiversity for 𝛾1 =
0.049 with experimental time of 10 years using ODE 45
𝑵 𝟏 𝑵 𝟏𝒎 BG(%) 𝑵 𝟐 𝑵 𝟐𝒎 BL(% )
4.0000 4.0000 0 10.0000 10.0000 0
10.6819 10.7674 0.8010.6094 10.6048 0.04
17.3033 17.4700 0.9610.1236 10.1074 0.16
20.5151 20.6862 0.839.4452 9.4213 0.25
21.7063 21.8641 0.738.9593 8.9328 0.30
22.1823 22.3306 0.67 8.6637 8.6368 0.31
22.4060 22.5491 0.648.4908 8.4641 0.31
22.5238 22.6639 0.628.3900 8.3637 0.31
22.5896 22.7279 0.618.3312 8.3051 0.31
22.6273 22.7645 0.618.2967 8.2708 0.31
Table.2: Evaluating the extent of biodiversity for 𝛾1 =
0.0525 with experimental time of 10 years using ODE 45
numerical
𝑵 𝟏 𝑵 𝟏𝒎 𝑩𝑳(%) 𝑵 𝟐 𝑵 𝟐𝒎 BG(% )
4.0000 4.0000 0 10.0000 10.0000 0
10.6819 10.4703 1.98 10.6094 10.6206 0.11
17.3033 16.8882 2.4010.1236 10.1638 0.40
20.5151 20.0849 2.109.4452 9.5052 0.64
21.7063 21.3077 1.848.9593 9.0259 0.74
22.1823 21.8070 1.698.6637 8.7316 0.78
22.4060 22.0442 1.628.4908 8.5582 0.79
22.5238 22.1696 1.578.3900 8.4567 0.79
22.5896 22.2399 1.558.3312 8.3971 0.79
22.6273 22.2803 1.538.2967 8.3621 0.79
Table.3: Evaluating the extent of biodiversity for 𝛾1 =
0.055 with experimental time of 10 years using ODE 45
𝑵 𝟏 𝑵 𝟏𝒎 BL(%) 𝑵 𝟐 𝑵 𝟐𝒎 BG(% )
4.0000 4.0000 0 10.0000 10.0000 0
10.6819 10.2618 3.9310.6094 10.6317 0.21
17.3033 16.4754 4.7910.1236 10.2036 0.79
20.5151 19.6507 4.219.4452 9.5653 1.27
21.7063 20.9031 3.708.9593 9.0933 1.50
22.1823 21.4256 3.418.6637 8.8004 1.58
22.4060 21.6763 3.268.4908 8.6266 1.60
22.5238 21.8095 3.178.3900 8.5244 1.60
22.5896 21.8844 3.128.3312 8.4642 1.60
22.6273 21.9277 3.098.2967 8.4286 1.59
Table.4: Evaluating the extent of biodiversity for 𝛾1 =
0.0575 with experimental time of 10 years using ODE 45
𝑵 𝟏 𝑵 𝟏𝒎 BL(%) 𝑵 𝟐 𝑵 𝟐𝒎 BG(% )
4.0000 4.0000 0 10.0000 10.0000 0
10.6819 10.0566 5.8510.6094 10.6427 0.31
17.3033 16.0652 7.1610.1236 10.2432 1.18
20.5151 19.2130 6.35 9.4452 9.6254 1.91
21.7063 20.4927 5.59 8.9593 9.1613 2.26
22.1823 21.0378 5.16 8.6637 8.8701 2.38
22.4060 21.3021 4.93 8.4908 8.6961 2.41
22.5238 21.4434 4.80 8.3900 8.5932 2.42
22.5896 21.5231 4.72 8.3312 8.5323 2.41
22.6273 21.5693 4.68 8.2967 8.4962 2.40
Table.5: Evaluating the extent of biodiversity for 𝛾1 = 0.06
with experimental time of 10 years using ODE 45
𝑵 𝟏 𝑵 𝟏𝒎 BL(%) 𝑵 𝟐 𝑵 𝟐𝒎 BG(% )
4.0000 4.0000 0 10.0000 10.0000 0
10.6819 9.8545 7.75 10.6094 10.6536 0.42
17.3033 15.6578 9.51 10.1236 10.2825 1.57
20.5151 18.7722 8.50 9.4452 9.6857 2.55
21.7063 20.0764 7.51 8.9593 9.2299 3.02
22.1823 20.6436 6.94 8.6637 8.9408 3.20
22.4060 20.9216 6.63 8.4908 8.7666 3.25
22.5238 21.0711 6.45 8.3900 8.6631 3.25
22.5896 21.1558 6.35 8.3312 8.6015 3.25
22.6273 21.2050 6.29 8.2967 8.5649 3.23
Table.6: Evaluating the extent of biodiversity for 𝛾1 =
0.095 with experimental time of 10 years using ODE 45
𝑵 𝟏 𝑵 𝟏𝒎 BL(%) 𝑵 𝟐 𝑵 𝟐𝒎 BG(% )
4.0000 4.0000 0 10.0000 10.0000 0
10.6819 7.3483 31.21 10.6094 10.7915 1.72

3. International Journal of Advanced Engineering, Management and Science (IJAEMS) [Vol-4, Issue-10, Oct-2018]
https://dx.doi.org/10.22161/ijaems.4.10.4 ISSN: 2454-1311
www.ijaems.com Page | 739
17.3033 10.3835 39.99 10.1236 10.7873 6.56
20.5151 12.4577 39.28 9.4452 10.5173 11.35
21.7063 13.7002 36.88 8.9593 10.2380 14.27
22.1823 14.4254 34.97 8.6637 10.0218 15.68
22.4060 14.8564 33.70 8.4908 9.8706 16.25
22.5238 15.1193 32.87 8.3900 9.7695 16.44
22.5896 15.2828 32.35 8.3312 9.7030 16.47
22.6273 15.3860 32.00 8.2967 9.6598 16.43
Table.7: Evaluating the extent of biodiversity for 𝛾1 =
0.0975 with experimental time of 10 years using ODE 45
𝑵 𝟏 𝑵 𝟏𝒎 BL(%) 𝑵 𝟐 𝑵 𝟐𝒎 BG(% )
4.0000 4.0000 0 10.0000 10.0000 0
10.6819 7.1914 32.6810.6094 10.8004 1.80
17.3033 10.0462 41.94 10.1236 10.8196 6.88
20.5151 12.0143 41.44 9.4452 10.5739 11.95
21.7063 13.2174 39.11 8.9593 10.3112 15.09
22.1823 13.9344 37.18 8.6637 10.1041 16.63
22.4060 14.3678 35.88 8.4908 9.9574 17.27
22.5238 14.6356 35.02 8.3900 9.8580 17.50
22.5896 14.8041 34.47 8.3312 9.7920 17.54
22.6273 14.9113 34.10 8.2967 9.7487 17.50
Table.8: Evaluating the extent of biodiversity for 𝛾1 =
0.099 with experimental time of 10 years using ODE 45
𝑵 𝟏 𝑵 𝟏𝒎 BL(%) 𝑵 𝟐 𝑵 𝟐𝒎 BG(% )
4.0000 4.0000 0 10.0000 10.0000 0
10.6819 7.0987 33.5510.6094 10.8057 1.85
17.3033 9.8468 43.09 10.1236 10.8388 7.07
20.5151 11.7501 42.73 9.4452 10.6075 12.31
21.7063 12.9272 40.45 8.9593 10.3550 15.58
22.1823 13.6374 38.52 8.6637 10.1537 17.20
22.4060 14.0713 37.20 8.4908 10.0099 17.89
22.5238 14.3415 36.33 8.3900 9.9117 18.14
22.5896 14.5127 35.76 8.3312 9.8461 18.18
22.6273 14.6223 35.38 8.2967 9.8028 18.15
IV. DISCUSSION OF RESULTS.
With an inter-competition coefficient value of 𝛾1 = 0.049
and an experimental time of ten (10) years, the biodiversity
gain percentage value has maintained a maximum value of
0.96 and biodiversity loss maximum percentage value of
0.31 from the sixth month. As 𝛾1 increases to 0.0525, the
maximum biodiversity gain percentage value is 2.40occurs
at the second month while the maximum biodiversity loss
percentage value of 0.79 starts occurring from the seventh
month. At an 𝛾1 value of 0.055, the maximum biodiversity
gain percentage value is 4.79 and biodiversity loss
percentage value is 1.60. When the 𝛾1 value is 0.0575, the
maximum biodiversity gain percentage value is 7.16 while
the maximum biodiversity loss percentage value is 2.42. An
𝛾1 value of 0.06 results in a maximum biodiversity gain
percentage value of 9.51 at the second month while the
maximum biodiversity loss percentage value is 3.25
occurring at the seventh and eighth months. When 𝛾1 is
0.095, the maximum biodiversity gain percentage value is
39.99 and maximum biodiversity loss percentage value is
16.47. At 𝛾1 = 0.0975, the maximum biodiversity gain value
is 41.94 while maximum biodiversity loss value is 17.54
and when 𝛾1=0.099, maximum biodiversity gain percentage
value is 43.09 and maximum biodiversity loss percentage
value is maintained at 18.18. Predominantly a biodiversity
gain is predicted at the second month while a biodiversity
loss is predicted from the seventh month and higher.
V. CONCLUSION
The MATLAB ODE45 numerical scheme has been used to
predict biodiversity gain and biodiversity loss resulting
from inter-competition between two phytoplankton species
at increasing inter-competition coefficient levels. A
relatively higher inter-competition coefficient would result
in a relatively lower biodiversity gain.
REFERENCES
[1] Anderson D.M. (1989). Toxic algae blooms and red
tides: a global perspective, in: T. Okaichi, D.M.
Anderson, T. Nemoto (Eds.), Red Tides: Biology,
Environ. Sci. Toxicol., Elsevier, New York, pp. 11–
21.
[2] Atsu, J. U. & Ekaka-a, E. N.(2017). Modelling
intervention with respect to biodiversity loss: A case
study of forest resource biomass undergoing changing
length of growing season. International Journal of
Engineering, Management and Science. 3(9).
[3] Bandyopadhyaya M., Saha T., Pal R. (2008).
Deterministic and Stochastic analysis of a delayed
allelopathic phytoplankton model within fluctuating
environment, Elsevier, 2:958–970.
[4] Bandyopadhyaya M. (2006). Dynamical analysis of a
allelopathic phytoplankton model, J. Biol. Sys. 14:
205–218.
[5] Chattopadhyay J., Sarkar R.R., Mondal S. (2002).
Toxin-producing phytoplankton may act as a
biological control for planktonic blooms-field study
and mathematical modeling, J. Theor. Biol. 215: 333–
344.

4. International Journal of Advanced Engineering, Management and Science (IJAEMS) [Vol-4, Issue-10, Oct-2018]
https://dx.doi.org/10.22161/ijaems.4.10.4 ISSN: 2454-1311
www.ijaems.com Page | 740
[6] Chattopadhyay J. (1996). Effects of toxic substances
on a two-species competitive system, Ecol. Model. 84:
287–289.
[7] Common, M., Perrings, C. (1992). Towards an
ecological economics of sustainability. Ecol.
Economics 6, 7–34.
[8] Duinker J., Wefer G (1994). Das CO2 und die rolle des
ozeans, Naturwissenschahten, 81:237–242.
[9] Ekaka-a E.N (2009). Computational and mathematical
modelling of plant species interactions in a harsh
climate. Ph.D Thesis, Department of Mathematics,
The University of Liverpool and The University of
Chester, United Kingdom, 2009.
[10] Hallegraeff G.M. (1993). A review of harmful algae
blooms and the apparent global increase, Phycologia
32:79–99.
[11] Hernándes-Bermejo, B., Fairén, V., (1995). Lotka-
Volterra representation of general nonlinear
systems.Math. Biosci. 140: 1–32.
[12] Loreau, M., (2000). Biodiversity and ecosystem
functioning: recent theoretical advances. Oicos 91(1),
3–17.
[13] May R.M. (2001). Stability and Complexity in Model
Ecosystems, Princeton University Press, New Jercy.
[14] Maynard-Smith J. (1974), Models in Ecology,
Cambridge University Press, Cambridge.
[15] Nisbet R.M., Gurney W.S.C. (1982). Modelling
Fluctuating Populations, Wiley Interscience, New
York.
[16] Perrings, C., (1995). Ecological resilience in the
sustainability of economic development. Economie
Appliquée 48(2), 121–142.
[17] Pykh Yu, A. (2002). Lyapunov functions as a measure
of biodiversity:theoretical background, Ecological
Indicators 2: 123–133.
[18] Rice E. (1984). Allelopathy, Academic Press, New
York.
[19] Saha, T. & Bandyopadhyay, M. (2009). Dynamical
analysis of toxin producing phytoplankton
interactions. Nonlinear Analysis, Real World
Applications 10: 314-332.
[20] Sarkar R.R., Chattopadhyay J. (2003). The role of
environmental stochasticity in a toxic phytoplankton-
non-toxic phytoplankton-zooplankton system,
Environmetrics, 14: 775–792.
[21] Smayda T. (1990). Novel and nuisance phytoplankton
blooms in the sea: Evidance for a global epidemic, In:
E. Graneli, B. Sundstrom, L. Edler, D.M. Anderson
(Eds.), Toxic Marine Phytoplankton, Elsevier, New
York , pp. 29–40.
[22] Solé J., García-Ladona E., Ruardij P., Estrada M.
(2005). Modelling allelopathy among marine algae,
Ecol. Model. 183:373–384.
[23] Tapaswi P.K., Mukhopadhyay A. (1999). Effects of
environmental fluctuation on plankton allelopathy, J.
Math. Biol. 39: 39–58